Superlinear elliptic inequalities on weighted graphs

Let $(V,μ)$ be an infinite, connected, locally finite weighted graph. We study the problem of existence or non-existence of positive solutions to a semi-linear elliptic inequality \begin{equation*} Δu+u^σ\leq0\quad \text{in}\,\,V, \end{equation*} where $Δ$ is the standard graph Laplacian on $V$ and $σ>0$. For $σ\in(0,1]$, the inequality admits no nontrivial positive solution. For $σ>1$, assuming condition \textbf{($p_0$)} on $(V,μ)$, we obtain a sharp condition for nonexistence of positive solutions in terms of the volume growth of the graph, that is \begin{equation*} μ(o,n)\lesssim n^{\frac{2σ}{σ-1}}(\ln n)^{\frac{1}{σ-1}} \end{equation*} for some $o\in V$ and all large enough $n$. For any $\varepsilon>0$, we can construct an example on a homogeneous tree $\mathbb T_N$ with $μ(o,n)\approx n^{\frac{2σ}{σ-1}}(\ln n)^{\frac{1}{σ-1}+\varepsilon}$, and a solution to the inequality on $(\mathbb T_N,μ)$ to illustrate the sharpness of $\frac{2σ}{σ-1}$ and $\frac{1}{σ-1}$.

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